What is$\cot ^{-1}9+\>\mathrm{cosec}\>{}^{-1}\left(\frac{\sqrt{41}}{4}\right)$

How many values of $\theta $, where$- \pi < \theta < \pi$ satisfy both the equations $\cot \theta =-\sqrt{3}$ and $\csc \theta =-2$ simultaneously?

If $x+\frac{1}{x}=2\cos \theta$ then what is $x^{3}+\frac{1}{x^{3}}$ equal to?

If $0\le x\le \frac{\pi }{2}$ then what is the number of values of x satisfying the equation $\tan x+\sec x=2\cos x$